Loading an Engine With a Ramp
Two mechanics must slide a 420 lb engine crate from the ground into a truck bed that is 2.5 ft high. They have a 12 ft steel ramp and a come-along rated at 1,000 lb. The crate base is steel and the ramp is steel plate; the coefficient of friction between them is about 0.25, and the ramp's own weight is negligible. The shop rule is that one person may push a load only if the sustained force stays at or below 60 lb. A 16 ft ramp is also available in the yard. The crew wants to know the real force required, whether a single person can do the job, and whether the longer ramp is worth the trouble.
1. Ignoring friction, find the ramp angle and the force the ramp multiplies the load down to.
Read the worked answer
Angle: sin θ = rise / ramp length = 2.5 / 12 = 0.2083 → θ ≈ 12.0°
Ideal mechanical advantage: MA = ramp length / rise = 12 / 2.5 = 4.8
Ideal effort: 420 lb ÷ 4.8 = 87.5 lb
So even in the frictionless case, the push exceeds the 60 lb one-person limit. Friction will only make it worse, so the ideal number is a lower bound, not the answer.
2. Include friction and determine the actual push force, the already-per-person force for two workers, and whether the 60 lb rule is met.
Read the worked answer
The normal force on an incline is W cos θ, not W.
- cos 12.0° ≈ 0.978 → normal force = 420 × 0.978 ≈ 410.8 lb
- Friction = μN = 0.25 × 410.8 ≈ 102.7 lb, acting down the ramp
- Push component = 87.5 lb
Total push = 87.5 + 102.7 ≈ 190 lb
Two workers share it: ≈ 95 lb each - still above the 60 lb limit, so the job needs either three people, the come-along, or a gentler slope. The 1,000 lb come-along is well within its rating (about 5× margin), so using it is the safest call.
3. If they switch to the 16 ft ramp, recompute the push force and judge whether the swap is worth it.
Read the worked answer
- sin θ = 2.5/16 = 0.1563 → θ ≈ 9.0°, cos θ ≈ 0.9877
- Ideal MA = 16/2.5 = 6.4 → push component = 420/6.4 ≈ 65.6 lb
- Friction = 0.25 × 420 × 0.9877 ≈ 103.7 lb
- Total ≈ 169 lb (about 85 lb each with two people)
Judgment: lengthening the ramp by 33% cut the total force only about 11%, from 190 lb to 169 lb. Friction depends on the weight and barely on the angle, so it dominates at shallow slopes and the ramp's mechanical advantage buys less and less. The longer ramp still does not reach 60 lb per person. Recommendation: use the come-along, or reduce friction ( rollers, soap, a smoother plate) rather than chasing a longer ramp.